Internal coordinate length

Reaction is assumed to have occurred if a particular internal coordinate q, such as a bond length, attains a... 

The second method for representing a molecule in 3D space is to use internal coordinates such as bond lengths, bond angles, and torsion angles. Internal coordinates describe the spatial arrangement of the atoms relative to each other. Figure 2-91 illustratc.s thi.s for 1,2-dichlorocthanc. 

Figure 2-91. Internal coordinates of 1,2-dichloroethane bond lengths and r2, bond angle a, and torsion angle r.

You need to specify two parameters the et uilibrium value ofthe internal coordinate and the force constant for the harmonic poten tial, T h e equilibrium restraint value deperi ds on the reason you choosea restraint. If, for example, you would like a particular bond length to remain constant during a simulation, then the equ ilibritirn restrain t value would probably be Lh e initial len gth of the bond. If you wan t to force an internal coordinate to a new value, the equilibrium internal coordinate is the new value. 

These difficulties have led to a revival of work on internal coordinate approaches, and to date several such techniques have been reported based on methods of rigid-body dynamics [8,19,34-37] and the Lagrange-Hamilton formalism [38-42]. These methods often have little in common in their analytical formulations, but they all may be reasonably referred to as internal coordinate molecular dynamics (ICMD) to underline their main distinction from conventional MD They all consider molecular motion in the space of generalized internal coordinates rather than in the usual Cartesian coordinate space. Their main goal is to compute long-duration macromolecular trajectories with acceptable accuracy but at a lower cost than Cartesian coordinate MD with bond length constraints. This task mrned out to be more complicated than it seemed initially. 

Figure 3 The underlying tree of a furanose ring in nucleic acids. Atoms are numbered 1,. . . , 5 corresponding to the natural tree ordering. All bond lengths are fixed. Arrows illustrate five internal coordinates that determine the ring conformation.

Over the years, geometry optimization has become an essential part of ab initio methodology. Research papers simply don t get published unless they report a geometry optimization. Almost all of the early ab initio packages made use of internal coordinates (bond lengths, bond angles and dihedral angles), as defined by the Z-matrix discussed in Chapter 1. The reason for the popularity of the... 

If tbe molecular geometry is optimized by the program, only a rough estimate of the parameters is necessary, hi term of internal coordinates, this is fairly easy. Some typical bond lengths (A) and angles are given below. 

FIGURE 12. Molecular internal coordinates (bond lengths and angles) to be combined in symmetry coordinates (see Table 2). 

Fig. 5.15 Schematic representation of the normal modes of the Fe(ni)-azide complex with the largest iron composition factors. The individual displacements of the Fe nucleus are depicted by a blue arrow. All vibrations except for V4 are characterized by a significant involvement of bond stretching and bending coordinates (red arrows and archlines), hi such a case, the length of the arrows and archlines roughly indicate the relative amplitude of bond stretching and bending, respectively. Internal coordinates vibrating in antiphase are denoted by inward and outward arrows respectively (taken from [63])...

As a simple illustration of the development of the secular determinant, consider the water molecule. A reasonable set of internal coordinates consists of the changes in lengths of the two bonds and the variation in the bond angle. Thus, from Eq. (45) and Fig. 2,... 

The QCRNA database is viewable and searchable with a web browser on the internet and it is also contained as a MySQL database that is easily incorporated with parameter optimization software to allow for the rapid development of specific reaction parameters. Molecular structures can be viewed with the JMOL [47, 48] or MOLDEN [49, 50] programs as viewers for chemical MIME types. If the web browser is JAVA-enabled, then the JMOL software will automatically load as a web applet. Both programs allow the structure to be manipulated, i.e., rotated, scaled, and translated, and allow for measurement of internal coordinates, e.g., bond lengths, angles, and dihedral angles. Similarly, animations of the vibrational frequencies are available and can be viewed with either program. 

The PES is only a function of the shape of the molecule, as described by the internal coordinates (there are ZN — 6 of them for a non-collinear molecule of N atoms). There are many possible choices of internal coordinates, including atom-atom distances (bond lengths), bond angles, dihedral or out-of-plane angles, or some combination of these. However, the PES can be expressed solely in terms of the atom-atom distances (a result that follows from the group theory of functions like the PES which are invariant to rotation).48,49 For N atoms there are NCi = N(N — l)/2 such distances, which are easily calculated from the Cartesian coordinates. Rather than use these atom-atom distances, Rn, we actually use the reciprocal distances, Zn ... 

Like many other chemical concepts the concept of strain is only semi-quantitative and lacks precise definition. Molecules are considered strained if they contain internal coordinates (interatomic distances (bond lengths, distances between non-bonded atoms), bond angles, torsion angles) which deviate from values regarded as normal and strain-free . For instance, the normal bond angle at the tetra-coordinated carbon atom is close to the tetrahedral value of 109.47°. In the course of force field calculations these normal values are defined more satisfactorily, though in a somewhat different way, as force field parameters. 

We describe as rigid-body rotation any molecular motion that leaves the centre of mass at rest, leaves the internal coordinates unaltered, but otherwise changes the positions of the atomic nuclei with respect to a reference frame. Whereas in a simple molecule, such as carbon monoxide, it is easy to visualize the two atoms vibrating about a mean position, i.e. with the bond length changing periodically, we may sometimes find it easier to see the vibration in our mind s eye if we think of one atom being stationary while the other atom moves relative to it. 

As is common with empirical force fields, MUBFF calculations are carried out using internal molecular coordinates rather than Cartesian coordinates. Internal coordinates describe the structure of a molecule in terms of bond lengths and angles between bonds. As an example, for a bent tri-atomic molecule ABC the three internal coordinates include the lengths of bonds AB (r s) and BC (rec), as well as the angle between them (aABc)- Larger molecules may also... 

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